Minimal Model Reasoning

One of the most successful form of non-monotonic reasoning is based on
the selection of minimal models. Among the various formalisms based on
minimal model semantics we consider here Circumscription
[41] and the Generalized Closed World Assumption (GCWA)
[42], which is a formalism to represent knowledge in a
closed world.

We assume that the reader is familiar with Circumscription, we briefly
present the definition of GCWA. The model semantics for GCWA is
defined as ( is a letter):

We can now present the results for these two formalisms.

Theorem 9
The problem of model checking for Circumscription is
coNP-complete, thus Circumscription is model-coNP-complete.

This result is a trivial corollary of a theorem already proved
[11, Theorem 6]. In fact, that proof implicitly
shows that model checking for circumscription is
coNP-complete.

Theorem 10
The problem of model checking for GCWA is in
P, thus GCWA is in model-P.

Proof. As already pointed out [11], it is possible
to rewrite into a propositional formula such that, for
any given model ,
if and only if .
Moreover, the size of is polynomially bounded by the size of .
As a consequence, the model compactness for GCWA is in the same
class of PL. By Theorem 3 the thesis
follows.

Theorem 11
The problem of inference for Circumscription is
-complete, thus Circumscription is thm--complete.

This result is a trivial corollary of a theorem published
in a previous paper
[11, Theorem 7] which implicitly
shows that inference for circumscription is
-complete.

Theorem 12
The problem of inference for GCWA is
coNP-complete, thus GCWA is
thm-coNP-complete.

Proof. As already pointed out in the proof of
Theorem 10, it is possible to rewrite
into a formula that is equivalent to it. As a consequence, a
formula is a theorem of if and only if it is a
theorem of . Thus, GCWA has at most the theorem compexity of PL.
Since GCWA is a generalization of PL, it follows that GCWA is in the
same theorem compactness class of PL. Hence, GCWA is
thm-coNP-complete.